❖♥ t❤❡ ◆✉♠❜❡r ❛♥❞ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❊①tr❡♠❡ P♦✐♥ts ♦❢ t❤❡ ❈♦r❡ ♦❢ ◆❡❝❡ss✐t② ▼❡❛s✉r❡s ♦♥ ❋✐♥✐t❡ ❙♣❛❝❡s ●❡♦r❣ ❙❝❤♦❧❧♠❡②❡r ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤ ✶✴✾ ●❡♦r❣ ❙❝❤♦❧❧♠❡②❡r ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤
❖✉r ❲♦r❦✐♥❣ ●r♦✉♣ ✷✴✾ ●❡♦r❣ ❙❝❤♦❧❧♠❡②❡r ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤
❇❛❝❦❣r♦✉♥❞✿ ❇❡❧✐❡❢ ❋✉♥❝t✐♦♥s ✭❙❤❛❢❡r✱ ✶✾✼✻✮ ❛♥❞ ◆❡❝❡ss✐t② ▼❡❛s✉r❡s ✭❉✉❜♦✐s✱ Pr❛❞❡✱ ✶✾✽✽✮ ◆❡❝❡ss✐t② ♠❡❛s✉r❡s ♦♥ ✜♥✐t❡ s♣❛❝❡s tr❡❛t❡❞ ❛s ❜❡❧✐❡❢ ❢✉♥❝t✐♦♥s✿ ❇❛s✐❝ ♣r♦❜❛❜✐❧✐t② ❛ss✐❣♥♠❡♥t ✭❜♣❛✮✿ m : ✷ Ω − → [ ✵ , ✶ ] ✇✐t❤ � m ( A ) = ✶✳ A ⊆ Ω ❆ss♦❝✐❛t❡❞ ❜❡❧✐❡❢ ❢✉♥❝t✐♦♥ Bel : ✷ Ω − → [ ✵ , ✶ ] : A �→ � m ( B ) ✳ B ⊆ A ❋♦❝❛❧ s❡ts ❛r❡ ❛❧❧ s❡ts A ⊆ Ω ✇✐t❤ m ( A ) > ✵✳ ❙❡t ♦❢ ❛❧❧ ❢♦❝❛❧ s❡ts ✐s ❞❡♥♦t❡❞ ✇✐t❤ F ( Bel ) ✳ ❆ ♥❡❝❡ss✐t② ♠❡❛s✉r❡ ✭♦♥ ❛ ✜♥✐t❡ s♣❛❝❡✮ ✐s ❛ ♠❛♣ N : ✷ Ω − → [ ✵ , ✶ ] s❛t✐s❢②✐♥❣✿ ∀ A , B ∈ ✷ Ω : N ( A ∩ B ) = ♠✐♥ { N ( A ) , N ( B ) } ✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t ✭♠❛t❤❡♠❛t✐❝❛❧❧②✱✮ ❛ ♥❡❝❡ss✐t② ♠❡❛s✉r❡ N ✐s ❛ s♣❡❝✐❛❧ ❜❡❧✐❡❢ ❢✉♥❝t✐♦♥✱ ♥❛♠❡❧② ❛ ❜❡❧✐❡❢ ❢✉♥❝t✐♦♥ ✇❤❡r❡ ❛❧❧ ❢♦❝❛❧ s❡ts ❛r❡ ♥❡st❡❞ ✭✐✳❡✳✿ ∀ A , B ∈ F ( N ) : A ⊆ B ♦r B ⊆ A ✮✳ ✸✴✾ ●❡♦r❣ ❙❝❤♦❧❧♠❡②❡r ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤
❚❤❡ ❈♦r❡ ♦❢ ❛ ❇❡❧✐❡❢ ❋✉♥❝t✐♦♥ ❖❜❥❡❝t ♦❢ ✐♥t❡r❡st✿ t❤❡ ❝♦r❡ ♦❢ Bel ✿ M ( Bel ) := { P ∈ P n | ∀ A ⊆ Ω : P ( A ) ≥ Bel ( A ) } ✳ M ( Bel ) ✐s ❛ ❝♦♥✈❡① ♣♦❧②t♦♣❡ t❤❛t ❝♦✉❧❞ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ✐ts ❡①tr❡♠❡ ♣♦✐♥ts ❡①t ( M ( Bel )) ✳ ❆✐♠✿ ❞❡s❝r✐❜✐♥❣ t❤❡ ❡①tr❡♠❡ ♣♦✐♥ts ♦❢ t❤❡ ❝♦r❡ ♦❢ ❛ ♥❡❝❡ss✐t② ♠❡❛s✉r❡ ✭♦r ♠♦r❡ ❣❡♥❡r❛❧✱ ♦❢ ❛ ❜❡❧✐❡❢ ❢✉♥❝t✐♦♥✮✳ ■♥ ❣❡♥❡r❛❧✱ ❞❡s❝r✐❜✐♥❣ t❤❡ ❡①tr❡♠❡ ♣♦✐♥ts ♦❢ t❤❡ ❝♦r❡ ♦❢ ❧♦✇❡r ♣r♦❜❛❜✐❧✐t✐❡s✴♣r❡✈✐s✐♦♥s ❝♦✉❧❞ ❜❡ ✉s❡❢✉❧ ❢♦r✿ ❞❡❝✐s✐♦♥ ♠❛❦✐♥❣ ✉♥❞❡r ♣❛rt✐❛❧ ♣r✐♦r ❦♥♦✇❧❡❞❣❡ st❛t✐st✐❝❛❧ ❤②♣♦t❤❡s✐s t❡st✐♥❣ ✉♥❞❡r ✐♠♣r❡❝✐s❡ ♣r♦❜❛❜✐❧✐st✐❝ ♠♦❞❡❧s ❞❡s❝r✐❜✐♥❣ t❤❡ ❝♦r❡ ♦❢ ❝♦♥✈❡① ❣❛♠❡s ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❣❛♠❡ t❤❡♦r② ✳ ✳ ✳ ✹✴✾ ●❡♦r❣ ❙❝❤♦❧❧♠❡②❡r ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤
b b b b ❚❤❡ ❝♦r❡ M ( Bel ) ✳✳✳ ✐s ♦❜t❛✐♥❡❞ ❛s ❛❧❧ ♠❛ss ♦❢ ❢♦❝❛❧ s❡ts A ✐s tr❛♥s❢❡rr❡❞ ❢r♦♠ t❤❡ ❢♦❝❛❧ s❡ts A t♦ ❡❧❡♠❡♥ts ω ∈ A ✭❈❤❛t❡❛✉♥❡✉❢✱ ❏❛✛r❛②✱ ✶✾✽✾✮✿ ∀ p ∈ M ( Bel ) ∃ λ : p ( { ω } ) = A ∋ ω λ A ( { ω } ) · m ( A ) � basic probability probability measure p mass transfer λ assignment m A 4 = { ω 2 , ω 3 , ω 4 , ω 5 } m ( A 4 ) = 0 . 4 A 3 = { ω 2 , ω 3 , ω 4 } m ( A 3 ) = 0 . 3 A 2 = { ω 4 , ω 5 } 1 1 = m ( A 2 ) = 0 . 2 = ) ) } } 2 3 ω ω { { A 1 = { ω 5 } ( 0 ( 3 . 3 A 4 . 7 A 0 λ λ m ( A 1 ) = 0 . 2 1 Ω ω 1 ω 2 ω 3 ω 4 ω 5 ✺✴✾ ●❡♦r❣ ❙❝❤♦❧❧♠❡②❡r ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤
b b b ❚❤❡ ❡①tr❡♠❡ ♣♦✐♥ts ♦❢ t❤❡ ❝♦r❡ s❛t✐s❢②✿ ■✮ ❆❧❧ ♠❛ss m ( A ) ✐s tr❛♥s❢❡rr❡❞ t♦ ❡①❛❝t❧② ♦♥❡ st❛t❡ ω ∈ A ✳ ■■✮ ■❢ ❛❧❧ ♠❛ss m ( A ) ✐s tr❛♥s❢❡rr❡❞ t♦ ω ❛♥❞ ❛❧❧ ♠❛ss m ( B ) ✐s tr❛♥s❢❡rr❡❞ t♦ ω ′ ❛♥❞ ✐❢ { ω, ω ′ } ⊆ A ∩ B t❤❡♥ ω = ω ′ ✳ I): II): B A A A ∩ B Ω Ω ω 2 ω 3 ω 4 ω 5 ω 1 ω 2 ω 3 ω 4 ω 5 ω 1 ✻✴✾ ●❡♦r❣ ❙❝❤♦❧❧♠❡②❡r ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤
b b b ❋♦r ❛ ♥❡❝❡ss✐t② ♠❡❛s✉r❡✳✳✳ ❚❤❡ ❡①tr❡♠❡ ♣♦✐♥ts ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❧♦♦❦✐♥❣ ❛t ❢♦❝❛❧ s❡ts ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ❝❛r❞✐♥❛❧✐t②✳ ❖❜s❡r✈❛t✐♦♥✿ ❚❤❡ ♠❛ss ♦❢ ❛ ❢♦❝❛❧ s❡t A k + ✶ ❝❛♥ ❜❡ tr❛♥s❢❡rr❡❞ ❡✐t❤❡r s♦♠❡✇❤❡r❡ ♦✉ts✐❞❡ t❤❡ ♣r❡✈✐♦✉s ❢♦❝❛❧ s❡t A k ✿ ✶ ❋♦r t❤✐s ♦♥❡ ❤❛s | A k + ✶ \ A k | ♣♦ss✐❜✐❧✐t✐❡s✳ ♦r s♦♠❡✇❤❡r❡ ✐♥t♦ t❤❡ ♣r❡✈✐♦✉s ❢♦❝❛❧ s❡t A k ✿ ✷ ❚❤❡♥ t❤❡ ♠❛ss ❤❛s t♦ ❜❡ tr❛♥s❢❡rr❡❞ t♦ t❤❡ s❛♠❡ ω t♦ ✇❤✐❝❤ t❤❡ ♠❛ss ♦❢ t❤❡ ♣r❡✈✐♦✉s ❢♦❝❛❧ s❡t A k ✐s tr❛♥s❢❡rr❡❞✳ 1 Ω ω 1 ω 2 ω 3 ω 4 ω 5 ✼✴✾ ●❡♦r❣ ❙❝❤♦❧❧♠❡②❡r ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤
b b b ❋♦r ❛ ♥❡❝❡ss✐t② ♠❡❛s✉r❡✳✳✳ ❚❤❡ ❡①tr❡♠❡ ♣♦✐♥ts ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❧♦♦❦✐♥❣ ❛t ❢♦❝❛❧ s❡ts ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ❝❛r❞✐♥❛❧✐t②✳ ❖❜s❡r✈❛t✐♦♥✿ ❚❤❡ ♠❛ss ♦❢ ❛ ❢♦❝❛❧ s❡t A k + ✶ ❝❛♥ ❜❡ tr❛♥s❢❡rr❡❞ ❡✐t❤❡r s♦♠❡✇❤❡r❡ ♦✉ts✐❞❡ t❤❡ ♣r❡✈✐♦✉s ❢♦❝❛❧ s❡t A k ✿ ✶ ❋♦r t❤✐s ♦♥❡ ❤❛s | A k + ✶ \ A k | ♣♦ss✐❜✐❧✐t✐❡s✳ ♦r s♦♠❡✇❤❡r❡ ✐♥t♦ t❤❡ ♣r❡✈✐♦✉s ❢♦❝❛❧ s❡t A k ✿ ✷ ❚❤❡♥ t❤❡ ♠❛ss ❤❛s t♦ ❜❡ tr❛♥s❢❡rr❡❞ t♦ t❤❡ s❛♠❡ ω t♦ ✇❤✐❝❤ t❤❡ ♠❛ss ♦❢ t❤❡ ♣r❡✈✐♦✉s ❢♦❝❛❧ s❡t A k ✐s tr❛♥s❢❡rr❡❞✳ A 3 A 2 A 1 1 Ω ω 1 ω 2 ω 3 ω 4 ω 5 ✼✴✾ ●❡♦r❣ ❙❝❤♦❧❧♠❡②❡r ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤
b b b ❋♦r ❛ ♥❡❝❡ss✐t② ♠❡❛s✉r❡✳✳✳ ❚❤❡ ❡①tr❡♠❡ ♣♦✐♥ts ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❧♦♦❦✐♥❣ ❛t ❢♦❝❛❧ s❡ts ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ❝❛r❞✐♥❛❧✐t②✳ ❖❜s❡r✈❛t✐♦♥✿ ❚❤❡ ♠❛ss ♦❢ ❛ ❢♦❝❛❧ s❡t A k + ✶ ❝❛♥ ❜❡ tr❛♥s❢❡rr❡❞ ❡✐t❤❡r s♦♠❡✇❤❡r❡ ♦✉ts✐❞❡ t❤❡ ♣r❡✈✐♦✉s ❢♦❝❛❧ s❡t A k ✿ ✶ ❋♦r t❤✐s ♦♥❡ ❤❛s | A k + ✶ \ A k | ♣♦ss✐❜✐❧✐t✐❡s✳ ♦r s♦♠❡✇❤❡r❡ ✐♥t♦ t❤❡ ♣r❡✈✐♦✉s ❢♦❝❛❧ s❡t A k ✿ ✷ ❚❤❡♥ t❤❡ ♠❛ss ❤❛s t♦ ❜❡ tr❛♥s❢❡rr❡❞ t♦ t❤❡ s❛♠❡ ω t♦ ✇❤✐❝❤ t❤❡ ♠❛ss ♦❢ t❤❡ ♣r❡✈✐♦✉s ❢♦❝❛❧ s❡t A k ✐s tr❛♥s❢❡rr❡❞✳ A 3 A 2 A 1 1 Ω ω 1 ω 2 ω 3 ω 4 ω 5 ✼✴✾ ●❡♦r❣ ❙❝❤♦❧❧♠❡②❡r ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤
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